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OPTICA ACTA, 1985, VOLi.32, NO. 3, 259 261 A depolarization criterion in Mueller matrices JOSE JORGE GIL Citedra de Fisica, Escuela Universitaria de Formaci6n Profesorado E.G.B.,. 26004 Logrofio, La Rioja, Spain and EUSEBIO BERNABEU Departamento de Optica, Facultad de Ciencias Fisicas, Universidad Complutense, 28040 Madrid, Spain (Received 14 December 1984; amended version received 11 March 1985) Abstract. A necessary and sufficient scalar condition for a Mueller matrix M to describe a non-depolarizing optical system is obtained. By computing only one scalar parameter, it can be determined whether a given Mueller matrix M describes a non-depolarizing, or a depolarizing optical system. The following theorem is stated: a necessary and sufficient condition for a given Mueller matrix M to describe a non-depolarizing optical system is Tr (MTM) = 4m2o. 1. Introduction The nature and mathematical expression of the relationships between the 16 elements of a Mueller matrix have been the subject of a number of previous papers [1, 2]. Abhyankar and Fymat [3] proved some relationships between the elements of the matrix that describes a non-depolarizing system in several matricial representations of polarization phenomena. Fry and Kattawar [4] performed a more detailed study of these relationships in the Stokes-Mueller formalism (SMF), which included some inequalities for depolarizing systems. A system of equations equivalent to the one given by the above authors [1,2,3], but with a different mathematical expression, was obtained by Barakat [5], and Simon [6] later obtained a matrix condition for a Mueller matrix, M, to describe a totally polarized system. In this paper we show that, because of the peculiar structure of Mueller matrices, which satisfy a series of constraining inequalities [2, 7], the set of nine scalar equalities involved in Simon's matrix condition can be replaced by only one set of scalar conditions, namely Tr(MTM)=4m00 . 2. Condition of the norm in Mueller matrices For every Mueller matrix M we can define a positive-semidefinite norm F(M), given by F(M) = [Tr (MTM)] /2 = E, m2] 2 (1) where mij(i,j= 0, 1, 2, 3) are the elements of M, Tr represents the trace, and MT is the transposed matrix of M. 260 Optica Acta Letters Fry and Kattawar [4] proved that for every non-depolarizing Mueller matrix M we can write 3 m2 = 4m02. (2) i,j=o This equation can also be written as F(M) = 2m 00 . (3) So, equation (3) is a necessary condition for M to describe a non-depolarizing system. However, we are now going to prove that equation (3) is also a sufficient condition. For an incident, totally polarized light beam, a depolarizing optical system produces an incoherent superposition of totally polarized outgoing light beams with different polarizations. Taking into account the principle of optical equivalence of polarization states [1], the above observation implies that a depolarizing optical system is optically equivalent to a system composed of a parallel combination of several non-depolarizing optical systems. Then, the outgoing light beam is optically equivalent to the one given by the superposition of the light beams emergent from each of the non-depolarizing optical systems of the equivalent parallel combination. The Mueller matrix of a parallel combination of optical systems is given by the sum of the Mueller matrices of the systems which form the parallel combination [1]. Therefore, a depolarizing Mueller matrix can be written as the sum of various nondepolarizing Mueller matrices. Since a depolarizing Mueller matrix depends, in general, on 16 independent parameters [1], and a non-depolarizing Mueller matrix depends, in general, on 7 independent parameters [1], we see that a depolarizing Mueller matrix can be written as the sum of at least 3 non-depolarizing Mueller matrices. Therefore we can write M=A+B+C+... (4) where A, B, C, . . , are different non-depolarizing Mueller matrices. They satisfy the following relations: Tr (ATA) = Tr (AAT) = 4ao 1 2 Tr (BTB) = Tr (BBT) = 4boo Tr (CTC) =Tr (CCT) = 4C2o0 ,. The decomposition given by equation (4) is not unique, but this fact does not affect our demonstration. For simplicity, we will consider the case in which M is obtained from only two matrices, A and B, so that M=A+B. (6) In order to prove that equation (3) is a sufficient condition, we assume that it is satisfied by M. Equation (6) allows us to write Tr (MTM) = Tr (ATA + ATB + BTA + BTB), (7) moo = ao + boo. (8) and 261 Optica Acta Letters Then 4moo = 4aoo + 4boo + 8aooboo. (9) Since Tr (ATB)= Tr (BTA), we can see that (10) Tr (MTM) = 4a2o0 + 4b2o + 2Tr (ATB). Thus, equations (2), (9) and (10) lead to (11) Tr (ATB) = Tr (BTA) = 4aooboo. From equations (5), and squaring in equation (11), we can write 16a2o0bo20 = Tr (ATA)Tr (BTB) = (Tr (BTA)) 2 . (12) However, taking into account the relation 3 Tr (BTA) = E aijbij, (13) i,j=O equation (2) becomes (ijoaij)( ijob)( aijbij). (14) This relation is only satisfied if A = B, so (15) M=2A=2B. In the general case of equation (3) we would have obtained A=B=C= ..... ,(16) Therefore, since A is a non-depolarizing Mueller matrix, and M is proportional to it, we can say that M is also non-depolarizing. We can summarize the above results with the following statement: a Mueller matrix M describes a non-depolarizing optical system if, and only if, it satisfies equation (3). This statement allows to to discover quickly, and through a simple criterion, whether or not a sample, whose Mueller matrix has been obtained experimentally, can produce depolarization of light. References H. C., 1957, Light Scattering by Small Particles(New York: Wiley). BASHARA, N. H., 1977, Ellipsometry and Polarized Light (Amsterdam: North Holland). ABHYANTAR, K. D., and FYMAT, A. L., 1969, J. Math. Phys., 10, 1935-1938. FRY, E. S., and KATTAWAR, G. W., 1981, Appl. Optics, 20, 2811-2114. BARAKAT, R., 1981, Optics Commun., 38, 159-161. SIMON, R., 1982, Optics Commun., 42, 293-297. SCHAEFER, R. W., Appl. Optics, 20, 2875. [1] VAN DE HULST, [2] [3] [4] [5] [6] [7] AZZAM, R. M. A., and